They are teaching alternating current in school. The book states that in an $RLC$ circuit, the instantaneous current is given as
$$i(t)=\frac{v_m}{\sqrt{R^2+(X_C-X_L)^2}}\sin(\omega t+\phi)$$
where $\tan\phi=\frac{X_C-X_L}R$, $v(t)=v_m\sin(\omega t)$. It's obvious from the equation that there is no damping of the current. The current is perfectly sinusoidal. This is also the case with RC circuit. Whereas, this is not the case in RL circuit, there is asymptotic growth in current.
Why is that?
To find more about the undergoing mathematical nuisance (the derivation in the book wasn't satisfactory enough for me, where we already assume that the current is sinusoidal), I tried Wolfram|Alpha. It seems like although an exponential factor exists, it has no effect asymptotically, except in the RL circuit where it leads to current asymptotically growing from $0$.
Then what's the problem?
What doesn't make sense to me is that we know for a fact that resistance dissipates energy, while pure inductor and pure capacitor don't. Then how it can be the case that the steady-state current is sinusoidal as the resistance will still be there in the circuit to dissipate energy. An "idea" I have for this is that maybe the inductor and capacitor work to store some amount of the energy (like in LC oscillations) and that portion, somehow, isn't affected by the resistance.
Please help me understand the undergoing mechanics. And if possible, how can we, without using Wolfram|Alpha, solve for the exact equations of the instantaneous current in RLC,LC circuits with alternating voltage? A reference to required material would be enough, as I want to do that on my own. I can solve it for the RL,RC circuits using integration factor method, but don't what to do for the others.
Please correct me if I got anything wrong.
Best Answer
You have left out something very important in presenting the problem: The source that provides power to the circuit.
If the source only provides power briefly (For example, during some interval $0 < t < T$), then you are right to expect the amplitude of the oscillation in the RLC circuit to decay for $t>T$ as the resistor dissipates the energy that has been provided by the source.
If the source provides power continuously (for example if you have a source voltage with the form $v_s=V\sin(\omega t + \phi)$ for all $t$, then the energy dissipated by the resistor can be replenished by the source, and you will find a steady-state solution that doesn't decay.
Presumably your book is discussing the second case when they found the result you are asking about.